[Math] Conditionally converges and rearrangement

real-analysissequences-and-series

Show that a conditionally convergent series has a rearrangement converging to $+\infty$

Thoughts:

  1. A conditionally convergent series is a series that converges but not absolutely converges $\lim_{m\to\infty} \sum_{n=0}^{m} a_n$ exists, $\sum_{n=0}^{\infty} |a_n|=\infty$

  2. if $\sum_{n=0}^{\infty} a_n$ is a conditionally convergent series, then for every real number $L$, there is a rearrangement that converges to $L$

Since we are given $\sum_{n=0}^{\infty} a_n$ converges conditionally. Intuitively, we obtain one positive and one negative series, which should be converge to the same limit.

Best Answer

This is a particular case of the astonishing Riemann Series Theorem . Note that you can rearrange a conditional convergent series in such a way as to make the rearranged series converge to whatever you want.

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